APPENDIX ONE: THE STANDARD CARTAN STRUCTURE EQUATIONS AND IDENTITIES USED IN ECE THEORY.

July 21, 2008
_{(Quotations from Evans' original text in} black, comments in blue)

. . .

The second structure equation of Cartan is:

                R^a = D Ù ω^a = d Ù ω^a + ω^a_cÙ ω^c                                       (1.5)

better written as

                R^a_b = d Ù ω^a_b + ω^a_cÙ ω^c_b                                                 (1.5')

. . .

The first identity of Cartan geometry is:

                DÙT^a := R^a_bÙ q^b                                                                 (1.7)

. . .

Finally the second identity of Cartan geometry is a restatement of the second structure equation:

                DÙR^a = DÙ(DÙ ω^a)                                                 (1.11)

better written as

                DÙR^a_b = DÙ(DÙω^a_b)                                                 (1.11')

Note that for a Christoffel connection, i.e. if and only if the torsion vanishes, Eq.(1.11) becomes

                DÙR^a_b = 0                                                                 (1.12)

Eq.(1.12) is always true, for general torsion, not only for a Christoffel connection. See e.g. F.W. Hehl's and Y.N. Obukhov's book [2, p.208] or G.W. Bruhn's web-note [3, Sect.2].